In order to find safe levels of environmental chemicals many toxicology experiments measure the effect of increasing doses of a substance on a group of animals. New methods will provide upper bounds on the effect of a toxin without making unwarranted assumptions about the shape of the dose response curve. The upper confidence bound on the effect of a given dose will be computed by finding the maximum value of X, where the likelihood ratio test accepts the hypothesis that the effect of the dose is grater than X. The likelihood ratio will be computed with the dose response curve constrained to be non-decreasing. Methods have preciously been developed to compute this value when there are a small number of distinct dose levels and no control group. The resulting confidence interval has advantages over the usual interval based on the standard error and over an interval based on a parametric assumption. The proposed reseach will develop methods for computing confidence bounds in other situations which are more common in practice. These are: when there is a zero dose control group; when the data is left censored, as in a survival study; when there are many dose levels, as in observational study; when there is a standard preparation for comparison, as in a proportional dilution bioassay; and when the data consists of a small number of binary responses as in a carcinogenesis study.